Nonhomogeneous Wavelet Systems in High Dimensions

It is of interest to study a wavelet system with a minimum number of generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in [11] that for any $d\times d$ real-valued expansive matrix M, a homogeneous orthonormal M-wavelet basis can be generated by a single wavelet function. On the other hand, it has been demonstrated in [21] that nonhomogeneous wavelet systems, though much less studied in the literature, play a fundamental role in wavelet analysis and naturally link many aspects of wavelet analysis together. In this paper, we are interested in nonhomogeneous wavelet systems in high dimensions with a minimum number of generators. As we shall see in this paper, a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system with almost all properties preserved. We also show that a nonredundant nonhomogeneous wavelet system is naturally connected to refinable structures and has a fixed number of wavelet generators. Consequently, it is often impossible for a nonhomogeneous orthonormal wavelet basis to have a single wavelet generator. However, for redundant nonhomogeneous wavelet systems, we show that for any $d\times d$ real-valued expansive matrix M, we can always construct a nonhomogeneous smooth tight M-wavelet frame in $L_2(R^d)$ with a single wavelet generator whose Fourier transform is a compactly supported $C^\infty$ function. Moreover, such nonhomogeneous tight wavelet frames are associated with filter banks and can be modified to achieve directionality in high dimensions. Our analysis of nonhomogeneous wavelet systems employs a notion of frequency-based nonhomogeneous wavelet systems in the distribution space. Such a notion allows us to separate the perfect reconstruction property of a wavelet system from its stability in function spaces

Wavelets and Wavelet Packets on Quantum Computers

We show how periodized wavelet packet transforms and periodized wavelet transforms can be implemented on a quantum computer. Surprisingly, we find that the implementation of wavelet packet transforms is less costly than the implementation of wavelet transforms on a quantum computer.Comment: 11 pages, 10 postscript figure, to appear in Proc. of Wavelet Applications in Signal and Image Processing VI

Wavelet design by means of multi-objective GAs for motor imagery EEG analysis

Wavelet-based analysis has been broadly used in the study of brain-computer interfaces (BCI), but in most cases these wavelet functions have not been designed taking into account the requirements of this field. In this study we propose a method to automatically generate wavelet-like functions by means of genetic algorithms. Results strongly indicate that it is possible to generate (evolve) wavelet functions that improve the classification accuracy compared to other well-known wavelets (e.g. Daubechies and Coiflets)

Design of Gm-C wavelet filter for on-line epileptic EEG detection

Copyright © 2019 The Institute of Electronics, Information and Communication EngineersAnalog filter implementation of continuous wavelet transform is considered as a promising technique for on-line spike detection applied in wearable electroencephalogram system. This Letter proposes a novel method to construct analog wavelet base for analog wavelet filter design, in which the mathematical approximation model in frequency domain is built as an optimization problem and the genetic algorithm is used to find the global optimum resolution. Also, the Gm-C filter structure based on LC ladder simulation is employed to synthesize the obtained analog wavelet base. The Marr wavelet filter is designed as an example using SMIC 1V 0.35μm CMOS technology. Simulation results show that the proposed method can give a stable analog wavelet filter with higher approximation accuracy and excellent circuit performance, which is well suited for the design of low-frequency low-power spike detector.Peer reviewe